PRCPrimeCalibrationForcesPrimeIdentityComparableTraceTarget_of_trace_coherence
plain-language theorem explainer
If prime calibration forces identity-orientation trace coherence on every ratio character, then it also forces the weaker comparable-trace respect property. Native-cost uniqueness and universal-foundation certificates cite this implication, as does the iff equating the two targets. The proof is a one-line pointwise lift of the character-level coherence-to-comparable implication.
Claim. Assume that every ratio-orbit map $\chi$ that is a ratio character and is prime-direction calibrated is identity-trace-coherent under prime calibration. Then every such $\chi$ also has the property that identity orientation respects comparability of finite $\delta$-orbit traces.
background
In the Primitive Recognition Calculus, a ratio character is a map $\chi$ on ratio orbits encoding how recognition orients multiplicative structure. Prime-direction calibration constrains how $\chi$ treats native prime axes. Two residual targets package what calibration should still force.
Trace coherence asks that identity orientation propagate across prime axes: once a prime axis is identity-oriented, related axes stay on the identity branch. The comparable-trace target is the sharper order-theoretic sibling: identity orientation must respect comparability of finite $\delta$-orbit traces. The character-level lemma already shows coherence implies comparable-trace respect (by dropping the unused comparability hypothesis in the conclusion).
This declaration lifts that character fact to the quantified calibration targets used by the native-cost uniqueness program.
proof idea
Term-mode, four lines. Introduce a ratio character $\chi$ together with the ratio-character and prime-direction-calibration hypotheses. Apply the assumed coherence target to those data to obtain PRCCharacterPrimeIdentityTraceCoherent χ. Feed that into PRCCharacterPrimeIdentityRespectsComparableTrace_of_trace_coherence, which returns the comparable-trace respect property for $\chi$. No extra algebraic work: pure pointwise specialization of the character-level implication.
why it matters
Closes one direction of the equivalence between the two residual prime-calibration targets: the iff theorem packages this arrow with its converse. That equivalence lets the uniqueness stack treat coherence and comparable-trace respect as interchangeable blockers.
Downstream, prc_native_cost_uniqueness_blocker_certificate and prc_universal_foundation_conditional_certificate sit on this layer of the PRC foundation. In the broader Recognition Science forcing chain, native cost uniqueness is the local step that pins the J-cost shape before T5 J-uniqueness and the RCL propagate to $\phi$, the eight-tick octave, and $D=3$. This lemma is bookkeeping, not physics, but it removes a fork in the residual target graph so certificates can cite a single Prop.
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